To find $ S_{ 38 } $ we use formula

$$ \color{blue}{S_n = \frac{n}{2} \cdot \left(2a_1 + (n-1) \cdot d \right) } $$

In this example we have $ a_1 = -30 ~,~ d = -10 ~,~ n = 38 $. After substituting these values into the above equation, we obtain:

$$ \begin{aligned} S_n &= \frac{n}{2} \cdot \left(2a_1 + (n-1) \cdot d \right) \\[1 em] S_{ 38 } &= \frac{ 38 }{2} \cdot \left( 2 \cdot \left( -30 \right) + ( 38-1) \cdot \left( -10 \right) \right) \\[1 em] S_{ 38 } &= \frac{ 38 }{2} \cdot \left( -60 + 37 \cdot \left( -10 \right) \right) \\[1 em] S_{ 38 } &= \frac{ 38 }{2} \cdot \left( -60 + \left( -370 \right) \right) \\[1 em] S_{ 38 } &= \frac{ 38 }{2} \cdot \left( -430 \right) \\[1 em] S_{ 38 } &= -8170 \end{aligned}$$

The first few terms of this sequence are:

$$ -30, ~~~-40, ~~~-50, ~~~-60, ~~~-70 . . . $$

Arithmetic sequences